20770_Problem_solving_Year_5_Number_and_place_value_Using_units_of_measurement_3

TEACHER NOTES
Problem-solving
To use logical reasoning and an ability to visualise a
sequence of events so as to then use number patterns to
solve problems
Materials
calculator
Focus
This page explores problems that involve a large amount
of data that needs both computation and patterning to
determine solutions.
Discussion
Page 61
These problems can be used to examine the ‘analyseexplore-try’
model of problem-solving that has evolved over
the varied number, spatial, measurement situations posed
in this book. This is discussed in detail in the introduction
pages (xiv – xvii). In this problem the information needs to
be carefully analysed to determine the distance of each
shell from the starting line, the number of shells that are
set out for the race and the distance to run to pick up
and return each shell to the bucket, being that only one
shell can be collected at a time. Some students may use
counters to represent the race and see what is happening.
Using a diagram is another way:
Another way is to look for a pattern:
4, 4 (1 + 2), 4 (1 + 2), 4 (1 + 2 + 3), 4 (1 + 2 + 3 + 4), …
The total must be 4 x the sum of all the numbers 1–50.
This result can be found by using a very famous method
supposedly first shown by a mathematician named Gauss
when he was only a young boy.
Sum wanted is 1 + 2 + 3 + 4 + 5 … + 50.
This sum is also 50 + 49 + 48 + … + 1.
Twice the sum must be 50 x 51, so the sum is half of 50 x
51. ([50 x 51 = 2500] ÷ 2 = 1275)
The winner must run 4 x 1275 = 5100 m.
This thinking now has to be applied in reverse to the
second problem. Since Jeff picked up 42 shells, the
distance he ran must be 4 x sum to 42 (i.e. 4 x half 42 x
43), which is 3612 metres; Jeff ran 3 km 612 m.
Possible difficulties
• Not understanding the way in which the shells are
distributed from the starting line
• Only using the distances to the shells
• Not attempting a solution because of the complexity
of the steps or calculations
Extension
• Use smaller or larger number of shells for the race
and vary the distance between shells.
• Ask students to research the mathematician Karl
Gauss and find out more about his famous solution to
the sum of the first 50 numbers.
When the problem is understood, a table that keeps track
of the distances to each shell and the cumulative distance
covered would enable an answer to be obtained, or even
allow students to see a pattern in the results that can be
applied to the whole situation.
Shell number Distance run Total distance (metres)
1 4 4
2 8 4 + 8
3 12 4 + 8 + 12
4 16 4 + 8 + 12 + 16
5 20 4 + 8 + 12 + 16 + 20
This would suggest that adding all the numbers using a
calculator which would give the correct answer of 5100 m
or more than 5 km (which at first seems surprising).
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Problem-solving in mathematics www.ricpublications.com.au R.I.C. Publications ®